WATER RESOURCES RESEARCH, VOL. 44, W01406, doi:10.1029/2007WR005944, 2008
Time domain particle tracking methods for simulating transport with retention and first-order transformation Scott Painter,1 Vladimir Cvetkovic,2 James Mancillas,1 and Osvaldo Pensado1 Received 7 February 2007; revised 12 September 2007; accepted 26 September 2007; published 3 January 2008.
[1] Particle tracking in the time domain has received increasing attention as a technique
for robustly simulating transport along one-dimensional subsurface pathways. Using a stochastic Lagrangian perspective, integral representations of transport including the effects of advection, longitudinal dispersion, and a broad class of retention models are derived; Monte Carlo sampling of that integral leads directly to new time domain particle tracking algorithms that represent a wide range of physical phenomena. Retention-time distributions are compiled for key retention models. An extension to accommodate linear transformations such as decay chains is also introduced. Detailed testing using first-order decay chains and four retention models (equilibrium sorption, limited diffusion, unlimited diffusion, and first-order kinetic sorption) demonstrate that the method is highly accurate. Simulations using flow fields produced by large-scale discrete-fracture network simulations, a transport problem that is difficult for conventional algorithms, demonstrate that the new algorithms are robust and highly efficient. Citation: Painter, S., V. Cvetkovic, J. Mancillas, and O. Pensado (2008), Time domain particle tracking methods for simulating transport with retention and first-order transformation, Water Resour. Res., 44, W01406, doi:10.1029/2007WR005944.
1. Introduction [2] Simulation of solute transport along one-dimensional (1-D) transport pathways is frequently required in hydrology applications. In discrete fracture networks, for example, transport is typically approximated as being 1-D between fracture intersections. In this situation, multiple segments representing individual fractures compose 1-D transport pathways that connect a source to a downstream monitoring location. When transverse dispersion can reasonably be neglected in continuous porous media (e.g., when all reactions are reasonably approximated as linear in the concentrations and the quantity of interest involves loworder spatial moments of flux at a monitoring boundary), a three-dimensional (3-D) transport problem can be decomposed into a series of 1-D transport problems by adopting a Lagrangian perspective [Dagan, 1982, 1984, 1989; Cvetkovic and Dagan, 1994; Cvetkovic et al., 1998]. In this case, each 1-D transport pathway is a streamline trajectory extracted from the 3-D flow field, an approach widely used in studies of potential low-level and high-level radioactive waste repositories. In all of these applications, computational efficiency is an important consideration, as the 1-D calculation is but one component of a larger simulation. [3] For 1-D pathways, particle tracking in the time domain has received increasing attention [Yamashita and Kimura, 1990; Moreno and Neretnieks, 1993; Robinson, 2000; Delay and Bodin, 2001; Tsang and Tsang, 2001; Reimus and James, 2002; Bodin et al., 2003]. Time domain 1 Center for Nuclear Waste Regulatory Analyses, Southwest Research Institute, San Antonio, Texas, USA. 2 Royal Institute of Technology, Stockholm, Sweden.
Copyright 2008 by the American Geophysical Union. 0043-1397/08/2007WR005944
particle methods differ from the more familiar particle tracking random walk [e.g., Tompson and Gelhar, 1990] method in that the latter uses a specified time step and a random spatial displacement, and the former uses a specified spatial displacement and a random transient time for the displacement. Time domain particle methods are appealing from a computational efficiency perspective because they allow particles to be moved across a constant-velocity segment of the pathway in a single step, which greatly decreases the computational requirements compared with space-based methods. In addition, matrix diffusion and other retention models are readily incorporated, and adaptive time stepping, which is required for efficient implementation of space-based methods and increases the complexity of those methods, is avoided completely. [4] Development of time domain particle methods is still at an early stage. Most of the previous work addressed a fairly narrow range of physical phenomena and was based on physical or heuristic arguments. The theoretical underpinnings of a general framework have not been fully illuminated. Longitudinal dispersion was either neglected completely or treated in an approximate manner. Although some preliminary work on transport of decay chains has been presented [Yamashita and Kimura, 1990], time domain particle methods have only recently been extended [Painter et al., 2006] to treat decay chains in a practical sense. [5] Recently, we introduced [Painter et al., 2006] a more general time domain particle method that represents longitudinal dispersion, matrix diffusion, and linear transformation processes such as radioactive decay and in-growth. The method is based on Monte Carlo sampling of residence times for particles, which represent packets of solute mass. The method was previously presented algorithmically and motivated by physical arguments. In this paper, we develop the method further, focusing on the theoretical justification
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for the method. Specifically, we generalize a well-known result from chromatography [Villermaux, 1987] for breakthrough in a system with transport, dispersion and retention. The new integral representation of the breakthrough curve includes a wide variety of retention processes and properly accounts for the hydrodynamic controls on the mass exchange between mobile and immobile regions; the Monte Carlo estimate of that integral representation of the breakthrough curve leads directly to the time domain particle tracking method. We also extended the algorithm to treat multiple species with linear transformations between species. In addition, several computational details are addressed including how longitudinal dispersion can be efficiently incorporated without resorting to approximations and how instantaneous breakthrough curves can be reconstructed from the particle arrival times.
2. Time-domain Particle Tracking Algorithm for a Single Species [6] We consider a 1-D pathway that connects a solute source to a monitoring location. The pathway may represent a Lagrangian trajectory/streamline or some other convenient idealization. The pathway is discretized into segments. In a discretely fractured rock application, a segment may represent that part of the pathway that passes through a single fracture. If an equivalent porous medium approach is used, a single segment may represent a hydrogeological unit. Transport and retention properties are allowed to vary from segment to segment. Velocity and mobile phase porosity (or fracture aperture, if the application involves fractured rock) are allowed to vary within each segment. Variation in physicochemical retention properties (e.g., distribution coefficients, immobile phase porosity) within a single segment is ignored here. However, some aspects of the formulation are left fairly general to facilitate future extensions of the method to spatially variable retention properties. [7] We are interested in mass discharge at a pathway (section) or segment outlet versus time. The mass discharge rate [mols/T] can be written as Zt rout ðt Þ ¼
ftran ðt t0 Þrin ðt 0 Þdt 0
ð1Þ
0
where rin is the rate at which mass is introduced into the pathway [mols/T]. The input rate can also be written as rin(t) = S0 fin(t) where S0 is the total source strength and fin(t) is the input rate normalized as a probability density. The kernel in the convolution, ftran, is the transit or residence time distribution for packets of mass in the pathway. The kernel may also be interpreted as the discharge rate due to a Dirac-d input. [8] A well-known result from chromatography [Villermaux, 1987] is to compute ftran as Z ftran ðttran Þ ¼
1
fret ðttran t Þ ft ðt Þ dt
ð2Þ
0
where ttran t = tret 0 is the retention time, which has probability density fret. In chromatography, this expression is usually written in the Laplace domain. Note also that in (2) the upper limit of integration extends to infinity instead of ttran because we include a Heaviside function centered at
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zero in our definition of the retention time distribution. The basic assumption in (2) is that the tracer concentration is relatively low (dilute systems) such that retention (mass transfer) processes are linear (in a generalized sense). In (2), t is the water residence time, ft is the water residence time density, and t represents the mean water residence time applicable to the considered pathway (section) or segment. The effect of longitudinal dispersion is accounted for by an appropriate choice of ft . An explicit form for ft(t) is discussed in Appendix A. [9] Equation (2) neglects variability in retention. It has been shown previously [Dagan and Cvetkovic, 1996; Cvetkovic et al., 1998, 1999; Cvetkovic and Haggerty, 2002] that diffusive transfer between mobile and immobile states, and hence retention, depends on velocity. Thus in heterogeneous media, retention is generally variable, similar to t, even when the chemical retention parameters are spatially constant. For a broad class of retention models, retention variability can be parameterized by a single velocity-dependent (random) parameter denoted b, and a set of uniform physicochemical retention parameters (factorized case, by Cvetkovic and Haggerty [2002]). Thus two velocity-dependent parameters characterize the transport: the nonretarded (water) residence time t and the transport resistance parameter b. The transport resistance parameter depends on the choice of retention model and will be discussed in section 3. For the case of matrix diffusion, b is the quantity 1/(b v) integrated along the pathway segment, where b is fracture half-aperture and v is fluid velocity. Regardless of the retention model, t and b are highly correlated through a shared dependence on the velocity field. In fact, if variability in mobile porosity (or fracture aperture) is neglected, b / t. [10] We generalize equation (2) to properly account for the coupling between retention and longitudinal dispersion by including longitudinal dispersion for both t and b Z
1
Z
1
ftran ðttran Þ ¼
fret ðttran tjb Þ fbjt ðbjt Þft ðt Þ dt db ð3Þ 0
0
where ft,b(t, b) = fbjt (bjt)ft(t) In other words, the t and b parameters become random variables with an appropriately specified joint density ft,b(t, b). This distribution depends on t and b, which are values for t and b in the absence of longitudinal dispersion and regarded as properties of the pathway section (segment), but this dependence is not specifically included in the notation. If spatial variability within each segment is neglected, tt = bb and fbjt ðbjt Þ ¼ d b tb=t
ð4Þ
where d is the Dirac-d function. More generally, the conditional probability density function (PDF) fbjt (bjt) can be generated by streamline tracing, as in Figure 4 of Cvetkovic et al. [1999]. [11] The mass discharge rate or breakthrough can now be written as Z
1
Z
1
Z
1
rout ðt Þ ¼ S0
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0
0
fbjt ðbjt Þft ðt Þ fin ðtin Þdt db dtin
ð5Þ
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and the cumulative tracer breakthrough curve as Z
Z
1
Rout ðt Þ ¼ S0
1
Z
1
Z
and algorithm verification tests for the transient velocity case are discussed by Painter et al. [2006].
1
H ðt tar Þ 0
0
0
3. Retention Time Distributions
0
fret ðtar tin t j b Þ fbjt ðbjt Þ
ft ðt Þ fin ðtin Þdt db dtin dtar
ð6Þ
where H() is the Heaviside function. Recall that our definition of retention time includes a Heaviside function at tret = 0, which is why the upper limits of integration are infinity in equations (5) and (6). A Monte Carlo estimate of equation (6) is X ^ out ðt Þ ¼ S0 H t tar;i R Npart i
ð7Þ
Here tar,i is one of Npart samples from the arrival time distribution, which has density Z
1
Z
1
Z
1
far ðtar Þ ¼
fret ðtar tin tjb Þ 0
0
0
fbjt ðbjt Þft ðt Þfin ðtin Þdtdbdtin
[22] It has been shown that the retention time distribution for a wide range of linear retention processes can be written in the Laplace domain as ~fret ðs; P1 ; P2 ; Þ ¼ expðsb~ g ðs; P1 ; P2 ; ÞÞ
1
Z
far ðtar Þ ¼
1
ð8Þ
fret tar tin tjtb=t ft ðt Þfin ðtin Þdtdtin ð9Þ 0
0
The Monte Carlo algorithm for the cumulative discharge curve is now assembled as follows: [12] (1) Sample a random start time tin from the normalized source fin(tin). [13] (2) Sample a t value as described in Appendix A. [14] (3) Sample a beta value from the conditional density fbjt (bjt). If internal variability is neglected, b = tb/t. [15] (4) Sample a retention time tret from fret(tretjb). [16] (5) Calculate the particle arrival time as tar = tin + t + tret. This value represents one sample from the arrival time distribution. [17] (6) For a given time t, if tar < t the particle contributes an amount S0/Npart to the cumulative mass discharge. [18] (7) Repeat from Step 1 a total of Npart times. [19] In practice, the arrival time and associated mass for each particle may be recorded and used in a postprocessing step to construct an approximation to Rout(t) and rout(t). Reconstruction of Rout(t) is simply a matter of summing the mass that arrives before a given time, as in step 6 and equation (7). Reconstruction of rout(t) is analogous to reconstructing probability density from a set of samples and is more difficult than the reconstruction of the Rout(t). A kernel method for reconstruction of rout(t) is discussed in Appendix B. [20] Steps (3) and (4) account for the interaction between longitudinal dispersion and retention simply and directly. Delay and Bodin [2001] and Bodin et al. [2003] recognize the need to address this interaction and propose an alternative algorithm. [21] The algorithm just described has been extended to include transient flow fields, albeit approximately. Details
ð10Þ
where s is the Laplace variable, b is the transport resistance parameter, g~(s) is the Laplace transform of the memory function g(t), and P1, P2, are physicochemical retention parameters, which are independent of the flow field. Note that this form is slightly different from that of Cvetkovic and Haggerty [2002] because we assume here that the physicochemical retention parameters are constant in the segment and incorporate the parameters into the memory function. The utility of the result is that it encapsulates all velocity dependence in the single parameter b. [23] In discretely fractured rock applications, Z b ðt Þ ¼
If internal variability is neglected, the corresponding distribution is Z
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0
t
d# bð#Þ
ð11Þ
where b is the fracture half-aperture and the integration is along the pathway/streamline. If aperture variability within each segment is ignored, then b = tb where b is the effective half-aperture for the segment. [24] Explicit definitions for b and the retention time distributions for four key retention models are given in Table 1. The equilibrium sorption, unlimited matrix diffusion, and limited matrix diffusion models are parameterized consistent with standard formulations. The two matrix diffusion retention models represent 1-D diffusion in the direction orthogonal to the fracture surface. In the unlimited diffusion model, diffusion is into an infinite half-space representing the matrix, which presumes the characteristic length scale for diffusion is small compared with the fracture spacing so that neighboring fractures can be ignored. In the limited diffusion model, diffusion is limited to a finite region immediately adjacent to the fracture surface. See Cvetkovic et al. [1999] for details. Several versions of the first order kinetic model exist. In Table 1, the formulation of Painter et al. [2001] is used with no retardation in the mobile zone. The governing partial differential equations in the Lagrangian framework are @C @C þ ¼ kf ðC N Þ @t @t
B
@N ¼ kf ðC N Þ @t
ð12aÞ
ð12bÞ
where C is mobile-zone concentration, N is immobile zone concentration, B = qqimm Rim and kf is a forward rate constant. An equivalent formulation is to parameterize with the backward rate constant k kf /B in place of the forward rate constant. The retention time PDF and cumulative distribu-
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Table 1. Retention Parameters for Various Retention Models Definition of Retention Transport Parameter Retention Model Resistance b Groupsa t R 1 Kdrb Equilibrium sorption qm ð#Þd# First order kinetic
0 Rt 0
1 qm ð#Þd#
qimRim, k
Retention Time CDF tret 0
Retention Time PDF d(tret bKdrb)
H(tret bKdrb)
ekmd(tret) + k2 mexp[k(m + tret)]^I 1[k2 mtret] where m bqimRim and ^I 1(Z) H(Z)I1(2Z1/2)/Z1/2
exp(mk p ffiffiffiffiffiffiffiffiffiffiffiffiffi ktret) {I0[2 k 2 mtret ] + y(ktret, km)} Ru pffiffiffiffiffi where y(u, v) exp(u) exp(x)I0(2 vx)dx 0
Unlimited Matrix Diffusion Limited matrix diffusion
Rt 0 Rt 0
1 bð#Þd# 1 bð#Þd#
rb = bulk density [M/L3] k = kinetic rate constant [1/T] H = Heaviside function qm = mobile porosity []
h 2 2i pffiffiffiffiffiffiffiffiffiffi ðtret Þkb b k qim DRim H p ffiffi 3=2 exp k 4tret 2 ptret qffiffiffiffiffi k, w D RDim L1(~ g (s)) ½2w1 where ln(~ g (s)) bkexp exp½2wþ1 qim = immobile (matrix) porosity [] D = matrix diffusion coefficient [L/T2] Kd = distribution coefficient [L3/M] L1(g) = Inverse Laplace transform of g s = Laplace variable
erfc[2pkbffiffiffiffi ] tret L-1(g~ðssÞ) Rim = immobile (matrix) retardation factor [] D = extent of matrix accessible by diffusion [L] I1 = Bessel function of the first kind of order 1 I0 = Bessel function of the first kind of order 0 erfc = Complementary cumulative error function
a
Presumed constant within each segment, but may vary from segment to segment and be different for each element in the decay chain.
tion function (CDF) in Table 1 are based on the latter formulation with spatially constant k in the segment.
4. Incorporating First-order Transformation Processes [25] Many contaminant transport applications involve multiple species with first-order transformations between species. Well-known examples include radionuclide decay chains and hydrocarbon degradation sequences (which are often approximated as first-order decay chains when the concentrations are low). Colloid-facilitated transport with reversible kinetically controlled transfer between colloids and solution is another example. [26] Yamashita and Kimura [1990] present a time domain algorithm that includes decay chains. Their method was developed for a single constant-property segment and a source that is constant over some specified release period. It is not clear how their algorithm could be extended to accommodate multiple segments and arbitrary source histories. [27] The complicating factor in transport analysis of radionuclide chains is that sorption and other physicochemical retention parameters are, in general, different for different members of the chain, which causes the retention-time distributions to be different. In our extension to the time domain particle tracking algorithm, decay and the resulting transformation to the next species in the decay chain are simulated as random events. If a decay event occurs in a segment, the total residence time in the segment is calculated as a weighted average of sampled residence times for the parent and offspring species. Note decay is treated as a random event that transforms the particle’s entire mass to the offspring species; the particle’s mass does not change until/unless the decay event occurs. [28] To be more specific, suppose that a particle enters a segment as species A. Species A decays to species B according to the first-order decay law with decay constant l. A decay time td = lnl R1 is first sampled for species A,
where R1 is an independent random number between 0 and 1. Two residence times denoted tA and tB for species A and B, respectively, are then sampled. (The residence time is the sum of the groundwater traveltime and the retention time, as described in section 2.) If td > tA, the particle survives the segment as species A, the particle’s clock is advanced by tA, and the algorithm proceeds to the next segment. [29] If td tA, the particle decays in the segment, and the clock is advanced by the amount td + (1 ttAd )tB. The first term in this expression represents the time in the segment spent as species A, and the second term represents the time spent as species B. [30] An important detail to note is that the sampled fullsegment residence times tA and tB must be perfectly correlated for the algorithm to work properly. Algorithmically, perfect correlation is enforced by using the same random number when generating a sample of tA and tB. It is easy to show that perfect correlation is required for the special case when A and B have identical sorption properties. Numerical experiments confirm this requirement in general; sampling the residence times for the parent and offspring species independently results in censoring of the residence-time distribution for any retention model with strong kinetic controls, thereby shifting the breakthrough curves to earlier times. The physical reason for this censoring is that a new sampling implies starting the offspring nuclide in the mobile fluid, whereas the parent nuclide is much more likely to be in the immobilized state (i.e., somewhere in the matrix) when the decay event occurs. [31] For clarity, the algorithm just described is for a twomember chain. The algorithm can be applied recursively, thus allowing decay chains of arbitrary length and multiple decay events in a single segment.
5. Verification of the Algorithm [32] The particle tracking algorithms described in section 2 have been implemented in two computer codes: TDRW and MARFA. Both codes also implement the first-order
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Figure 1. Benchmark results for a three-member decay chain with advection, longitudinal dispersion, and equilibrium sorption. The individual data points represent mass discharge calculated by the TDRW software using the methods introduced here. The continuous curves are from the GoldSim [GoldSim Technology Group, LLC, 2005] software. TM
transformation algorithms described in section 4. TDRW uses a deterministic representation of the pathway. MARFA has options for deterministic or stochastic representation of the transport pathway and is designed to work directly with output from discrete fracture network flow simulations. Tests of the algorithm and its implementation are presented in this section. [33] Instantaneous discharge (breakthrough) using the equilibrium sorption model and a three-member decay chain A ! B ! C is shown in Figure 1. This test used the TDRW code to simulate transport on a 20 m segment with a velocity of 1 m/a and a dispersivity of 1 m. Half-lives are 700, 1000, and 10,000 a, and the dimensionless distribution coefficients are 500, 100, and 500 for species A, B, and C, respectively. Radionuclide mass was instantaneously injected into the upstream end of the pathway as species A. Data points represent the normalized breakthrough curves (mass discharge at the pathway outlet versus time and normalized by the total injected mass) from the TDRW code with 105 particles, and thick curves are results of a purely numerical simulation using the commercial software GoldSim (registered trademark of GoldSim Technology Group, LLC [2005]). The TDRW and GoldSim results are very close and overplot each other over much of the range. Small differences at the leading edge are due to statistical fluctuations caused by the limited number of particles. Tests with other combinations of dimensionless distribution coefficients also yielded excellent agreement. [34] Validation tests for the unlimited matrix diffusion model used a two-member decay chain A ! B. The governing equations for this system are @Ci @Ci @ 2 Ci qD @Mi þv ajvj 2 ¼ li Ci þ li1 Ci1 bð xÞ @z z¼0 @t @x @x
Rim;i
@Mi @ 2 Mi ¼ D 2 li Rim;i Mi þ li1 Rim;i1 Mi1 @t @z
ð13aÞ
ð13bÞ
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for i = 1,2 and with boundary/initial conditions Ci(t, x) = Mi(t, x, 0), Mi(t, x, 1) = 0, C1(t, 0) = f(t), C2(t, 0) = 0, and Ci(0, x) = 0. Here Ci(t, x) is the concentration in the mobile pathway (fracture) for the i-th member of the chain, Mi(t, x, z) is the concentration in the matrix, x is distance in the direction of the pathway, z is distance orthogonal to the pathway, v is velocity, a is dispersivity, l is decay constant, and C0 0. The system was solved by the method of lines. That is, the x- and z-directions were discretized using blockcentered finite differences. That discretization procedure eliminated the x- and z-dependences and resulted in a large system of ordinary differential equations to be solved for the cell concentrations versus time. The ordinary differential equations were solved using commercial software. [35] Instantaneous discharge (breakthrough) curves for the matrix-diffusion validation test are shown on a linear scale in Figure 2 and on a log-log scale in the insert to Figure 2. The pathway is 100 m long, and the speciesindependent parameters are v = 1 m/a, a = 0.5 m, b = 0.1 mm, and D = 106 m2/a. The matrix retardation factors are 1000 and 10,000 for species A and B, respectively. Half-life is 10,000 a for both species. The source strength is initially 0.001 mol/a and decreases exponentially with decay constant of 0.001 a1. As in Figure 1, the TDRW solution and the target benchmark solution overplot each other over a wide range and show discernable differences only at the extreme leading edge of the breakthrough curve. The TDRW results are uncertain at the leading edge because a finite number of particles are used. The numerical results are also suspect at the leading edge because of the finite difference approximation used to eliminate the spatial variables x and z. Other tests were performed using different combinations for parameters (not shown), and all of these resulted in similar good agreement. [36] Verification tests for the limited matrix diffusion model also used a two-member decay chain A ! B. The governing equations are the same as equation (13) except the boundary condition Mi(t, x, 1) = 0 is replaced with @Mi @z z¼D = 0, where D = 2 mm. Other parameters are the
Figure 2. Benchmark results for a two-member decay chain with advection, longitudinal dispersion, and unlimited matrix diffusion. The individual data points represent mass discharge calculated by the TDRW software using the methods introduced here. The continuous curves are from an independent numerical simulation. The discharge curves are shown on a double logarithmic scale in the insert.
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benchmark over the entire range. We note also that this test is one of the simulations where the adaptive Gaussian kernel method produced poor reconstruction of the discharge, as discussed in Appendix B. In particular, the Gaussian kernel produced nonzero discharge at zero (and negative) times. The lognormal kernel produced no such artifacts, as can be seen in Figure 4.
6. Example Using Discrete Fracture Network (DFN) Trajectories
Figure 3. Benchmark results for a two-member decay chain with advection, longitudinal dispersion, and limited matrix diffusion. The individual data points represent mass discharge calculated by the MARFA software using the methods introduced here. The continuous curves are from an independent numerical simulation, as described in the text.
same as in the unlimited diffusion test except Rim is 200 for species A and 500 for species B. Results of the limited diffusion simulations are shown in Figure 3. The Monte Carlo results agree very well with the benchmark solution over the entire range. [37] Verification tests for the first-order kinetic model used a two-member decay chain and a 20 m pathway with a velocity of 1 m/a and a dispersivity of 1 m. Both species had a half-life of 1000 a and a forward rate constant of 0.1 a1. The transport resistance parameter b was 100 for species A and 300 for species B. The radionuclide source is identical to the source in the matrix diffusion tests. Results of the TDRW code are compared with numerical results obtained by finite difference solution of the governing equations in Figure 4. The TDRW results agree very well with the
Figure 4. Benchmark results for a two-member decay chain with advection, longitudinal dispersion, and firstorder kinetic sorption (mobile-immobile model). The individual data points represent mass discharge calculated by the TDRW software using the methods introduced here. The continuous curves are from an independent numerical simulation, as described in the text.
[38] The MARFA code has been used to compute radionuclide decay chain transport in a number of realistic examples that use trajectories extracted from detailed DFN simulations. The example described in this section is based on the DFN simulations described by Cvetkovic et al. [2004]. Those simulations used 20,000 simulated fractures in a 100 m 100 m 100 m region and an applied hydraulic gradient of 0.001 m/m. A total of 806 advectiononly trajectories representing equally probably pathways were extracted from the flow simulations and input into the MARFA code. The trajectories were generated by releasing particles on the upstream face and tracking those particles advectively through the network. The probability for selecting one of the fractures on the upstream face was proportional to flow in that fracture (flux-weighed release). At each fracture intersection, the advecting particle was assigned randomly to the downstream fractures with probability proportional to the outgoing flux (perfect mixing assumption). The average number of fracture segments per trajectory is about 35. Figure 5 shows 20 of the pathways that were selected randomly from the full set.
Figure 5. Example transport pathways extracted from a DFN simulation with 20,000 fractures. The pathways were determined by advective particle tracking with mixing at fracture intersections. The examples shown represent only a small fraction of the complete set of 806 pathways. The complete set was used in constructing Figures 6 and 7.
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Table 2. Half-Lives and Retention Parameters for the Decay Chain Used in the DFN Example
Radionuclide 241
Am Np 233 U 229 Th 237
Half-Life [a] 432 2.14 106 1.58 105 7339
Retention Parameter k [m/a1/2] 6.0 8.0 4.0 3.7
104 104 104 104
Matrix Thickness Parameter qffiffiffiffiffi D RDim [a1/2] 2.3 2.6 1.9 1.1
103 103 103 103
[39] The decay chain 241Am ! 237Np ! 233U ! 229Th was used. Mass was released into the pathways as 241Am. The total amount released into all pathways was 1 mole. Note that this is not meant to be a representative release scenario. It is intended purely as a test of the MARFA code to simulate realistic transport scenarios. Two retention models were used: the unlimited diffusion model and the limited diffusion model with an accessible matrix region of 20 mm thickness. Half-lives and retention properties are shown in Table 2. The matrix thickness parameter is used in the limited diffusion model only. The matrix thickness is approximated as infinite in the unlimited diffusion model. [40] Results for the infinite diffusion model are shown in Figure 6. For this example, about 76% of the initial mass passes through the 100 m system on a cumulative basis. The remaining mass decayed to one of the decay products of 229 Th, which are not tracked. Despite the relatively high cumulative throughput in this example, the mass release is spread over an extremely large time and the amount of mass released in any given year is a small fraction of the released mass (maximum annual discharge is about 103 % of the initial pulse). The large spreading in the discharge has two sources. The spreading is caused, in part, by the infinite diffusion model, which has strong kinetic controls. The spreading is also caused by a large variation in traveltime
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among the different pathways, which is a type of macrodispersion. Note in particular the early arrival time for a small fraction of the mass representing a small number of fast pathways in the DFN. [41] Results for the limited diffusion model are shown in Figure 7. Parameters are the same as in Figure 6 except that diffusion is allowed only in a 20 mm matrix region adjacent to the fracture. Peak discharge for the limited diffusion model is about 20 times that of the unlimited diffusion model and occurs at an earlier time. Nearly all of the initial mass passes through the 100 m block with the limited diffusion model. [42] To appreciate the computational efficiency of the method, note that the particle tracking and reconstruction steps in the large-scale simulations of Figures 6 and 7 took approximately 150 s of computing time for the particle tracking and reconstruction steps. Two broad classes of conventional methods could be used to solve the same problem. In the first conventional approach, a finite element grid would be imposed on each of the 20,000 fractures in the simulation. That approach would require simultaneous solution for a large number of nodal concentrations. The second conventional approach would be to extract the advective trajectories, as in this study, and then solve for concentration on each of the pathways using a conventional numerical code. Although clearly more tractable than full discretization, the second approach would still require 806 individual transport solutions to obtain the ensemble-average breakthrough. Clearly the time domain methods presented here are competitive with these conventional approaches. Part of that computational advantage comes from the fact that the time domain Monte Carlo approach does not need to resolve the transport in each of the 806 pathways. The nature of the Monte Carlo sampling in Figures 6 and 7 is such that the ensemble discharge may be accurately calculated even when the number of particles is too small to accurately resolve the transport in each of the 806 pathways.
7. Discussion and Conclusions [43] Solute transport simulations are needed in many applications, but studies of radionuclide transport place
Figure 6. Discharge versus time for a four-member decay chain transported through a discrete fracture network made up of 20,000 fractures. The curves are ensemble averages of 806 pathways extracted from the network by advective particle tracking. A total of 1 mole of radionuclides was introduced into the system as 241Am. Unlimited matrix diffusion is used as the retention model. Short-dashed curves use 5.0 105 particles, long-dashed curves use 106 particles, and solid curves use 5.0 106 particles.
Figure 7. Discharge versus time for the same scenario as Figure 5, but with the limited diffusion model and a 20 mm accessible matrix region.
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particular demands on the robustness and efficiency of simulation algorithms. Such studies require transport simulation for multiple species that are linked through decay chains and often have highly disparate sorption parameters. Monte Carlo approaches are commonly used with sampling of uncertain transport parameters. As a consequence, transport simulations are typically repeated many times and must accommodate a wide range of Peclet numbers. Matrix diffusion and other kinetically controlled retention processes must also be represented. If discrete fracture network simulations are used explicitly, the transport simulations must accommodate a great deal of spatial heterogeneity, which leads to large grids for finite difference or finite element methods. [44] The advantages of the time domain particle tracking approach over deterministic or space-based particle tracking have been recognized previously [e.g., Delay and Bodin, 2001]. The particular variant developed in this paper accommodates a much wider range of physical phenomena than previous versions. In particular, the algorithms described here accommodate decay chains, time-dependent velocity, and a variety of retention processes. These extensions were necessary to make the time domain approach practicable for large-scale applications such as nuclear waste repository evaluations. In addition, the algorithms presented here have a clear link with existing transport theory from subsurface hydrology and chemical engineering; longitudinal dispersion and the interaction between longitudinal dispersion and retention are represented without approximations. The numerical tests summarized in this paper demonstrate that the algorithm is extremely robust and highly accurate. Experience with the algorithm in largescale simulations demonstrates that it has clear advantages with respect to computationally efficiency, especially when the simulations include many finely discretized pathways. [45] The presented methodology is applicable to transport of tracers that are sufficiently dilute such that mass transfer between mobile and immobile states is linear in the generalized sense [Villermaux, 1987]; this implies that the ratio of Laplace transforms of the immobile and mobile concentrations is equal to the Laplace transform of the so-called memory function. This formulation can capture a wide range of kinetic behavior, primarily of the type where access to sorption sites is diffusion limited. The method is not applicable to situations where concentrations are high enough to make activity coefficients significantly different from unity or to cause kinetic rate parameters to become dependent on concentrations. The method also does not admit nonlinear empirical or mechanistic sorption models. Further, the method cannot treat general multicomponent reactive transport formulations wherein equilibrium mass action expressions replace a subset of the unknown concentrations because such formulations become nonlinear in the remaining concentrations [e.g., Lichtner, 1996; Steefel and MacQuarrie, 1996]. [46] Time domain approaches are suited to transport on 1-D pathways like those that occur naturally in discrete fracture network models. Classical transverse dispersion is not straightforwardly accommodated. For any transport scenario in which transverse dispersion can reasonably be neglected, a classical Lagrangian approach can be adopted, and the 3-D transport problem becomes equivalent to a
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series of 1-D transport problems solved along streamlines extracted from the 3-D flow field. The time domain algorithms described here are directly applicable in this case. Transverse dispersion is well understood to have only a minor effect if the solute source size is large, the quantity of interest involves integrated breakthrough at a monitoring boundary, or if a stochastic framework is adopted and the quantity of interest is expected concentration or flux [Fiori et al., 2002]. For highly localized sources or when calculating realization-to-realization variability of concentration in a stochastic framework, transverse dispersion is generally more important and space-based random walk or conventional numerical transport methods are better suited. [47] Although transverse dispersion is not straightforwardly represented, three-dimensional spreading may be represented in other ways for many applications. In discrete fracture networks, for example, transport within each fracture plane is often represented as 1-D because mixing or streamline routing at fracture intersections introduces threedimensional spreading that easily dominates any in-plane transverse dispersion. This process is represented in the simulations of Figures 5 and 6 through the calculation of the advective trajectories/streamlines that form the transport pathways. In addition, it may be possible to extend the time domain approach to approximate transverse dispersion. Indeed, some work along that line has already been published. Robinson [2000] used a ‘‘cell-based particle tracking’’ method to describe transport in the unsaturated zone beneath the potential repository at Yucca Mountain. That method samples a random residence time in each cell similar to the time domain methods described here and then redistributes particles randomly to downstream cells to approximate transverse dispersion. [48] The time domain particle methods described in this paper can be combined with stochastic pathway simulation algorithms described previously [Painter and Cvetkovic, 2005]. That combination was introduced recently [Painter et al., 2006] as the Particle On Random Streamline Segment (PORSS) algorithm. The motivation for the PORSS algorithm is to simulate radionuclide transport without invoking continuum-type assumptions at regional scales where DFN simulations would be infeasible. To accomplish this objective, the PORSS algorithm uses a pool of fracture segment properties extracted from relatively small DFN simulations. The segment pool is then used in the algorithms of Painter and Cvetkovic [2005] to construct artificial pathways that mimic, in the statistical sense, real pathways. Particle displacement in the PORSS algorithm is a two-step process: a pathway segment is first simulated, and then the particle is moved on that segment according to the algorithm introduced in this paper. The advantage of the approach is that the artificial pathways are not limited by the same size constraints as DFN simulations. As we will describe in a future publication, the PORSS algorithm can be combined with equivalent porous medium flow models to construct a multiscale algorithm for regional-scale simulation of radionuclide transport in discretely fractured rock.
Appendix A:
Longitudinal Dispersion
[49] As discussed in section 2, longitudinal dispersion can be incorporated by randomizing t and sampling from the appropriate distribution. The appropriate form for the
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density ft (t) and the associated sampling scheme are discussed in this section. [50] The density ft (t) has two interpretations. We can view it as a probability density for transit time of nonretarded tracers, as in section 2. Equivalently, it can be viewed as the mass breakthrough curve (monitored at a specified location) for the advection-dispersion equation with Dirac-d mass injection at the inlet of a semi-infinite system. Note that both injection and detection should be in terms of flux (not concentration) in this interpretation [see, e.g., Kreft and Zuber, 1978]. [51] Adopting the second interpretation, the following result is obtained from simple manipulation of existing solutions [Ogata and Banks, 1961] for the breakthrough curve ! rffiffiffiffiffiffi h 0 3=2 h ð1 t 0 Þ2 tft ðt Þ ¼ ðt Þ exp 4p 4 t0
ðA1Þ
where t 0 = tt , h = al , l is the length of the segment, a is the dispersivity, and t is the groundwater traveltime (a property of the segment). Bodin et al. [2003] note that this probability density can be approximated as lognormal for some ranges of h. [52] For sampling, itR is convenient to have the cumulative distribution Ft (t) t0 ft(s)ds instead of the density. The cumulative distribution can be obtained by direct integration of ft (t). Alternatively, it can be obtained from known analytical results after noting that the cumulative distribution is equivalent to the cumulative breakthrough curve for the advection-dispersion equation with a Dirac-d mass injection at the inlet of a semi-infinite system. A straightforward manipulation of breakthrough solutions from Kreft and Zuber [1978] yields 1 Ft ðt Þ ¼ erfc 2
pffiffiffi pffiffiffi h 1 t0 h 1 þ t0 1 pffiffiffiffi þ expðhÞerfc pffiffiffiffi ðA2Þ 2 2 2 t0 t0
This distribution can be sampled by first generating a random number R 2 [0, 1] and then solving for a sample value t * from Ft(t *) R. This equation can be solved by the Newton-Raphson method. However, some care must be exercised in the Newton-Raphson implementation to prevent the algorithm from encountering a very small value for the derivative of equation (A2), which can lead to spurious extrapolations and convergence failures. To prevent this, the Newton-Raphson method can be started pffiffiffiffiffiffiffiffiffi ffi 9þ3h2 3 0 , which is the location of the peak in at the t value h the derivative of equation (A2) with respect to t 0. Starting from this point ensures that the Newton-Raphson method is always moving in the direction of decreasing gradient, thus approaching the root from one side and avoiding spurious extrapolations. [53] Reimus and James [2002] presented an alternative t distribution based on an infinite-series representation. Their t distribution was developed by solving for the breakthrough curve using a Dirac-d function for the time-dependent concentration at the inlet and a zero-concentration boundary at the outlet. The Reimus and James [2002] distribution is correct for those boundary conditions, but provides only an approximate representation for use in the TDRW
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algorithm. If the TDRW algorithm is to be used to represent transport in a pathway made up of multiple segments, the typical scenario for an application, then the output of one segment forms the input to the next segment, which dictates that injection and detection in flux be used in deriving the t distribution. The importance of the choice of boundary conditions can be understood by noting that equation (A2) produces the same result if applied to N identical segments of length l in series or to a single segment of length Nl. The Reimus and James [2002] distribution and the lognormal approximation of Delay and Bodin [2001] do not have this property, although both approximate it.
Appendix B: Curves
Reconstruction of the Breakthrough
[54] The Monte Carlo algorithms produce output in the form of particle arrival times at pathway end points. Cumulative mass discharge (cumulative breakthrough) at a given time can be readily constructed from the arrival times by simply identifying the amount of mass arriving before the specified time. This procedure is equivalent to estimating cumulative probability distributions from a set of random samples. Estimating the mass discharge rate (instantaneous breakthrough) is more difficult. When the number of particles of each species surviving to the pathway endpoints is large, simple binning (histograms) may be sufficient for estimating breakthrough curves. However, the histogram does not make efficient use of the information contained in the arrival times. For the highly attenuating pathways that are often encountered in studies of potential nuclear waste repositories, the vast majority of released particles may decay before reaching the pathway endpoints; it is important in these applications to have methods that make efficient use of the arrival times of the surviving particles. Fortunately, the task of reconstructing breakthrough curves from the particle arrival times is analogous to the task of reconstructing probability density from a set of sampled values, a classical problem in statistics with a large literature [e.g., Silverman, 1986]. The optimal method is highly problem-dependent [Silverman, 1986] and typically requires some experimentation to identify. Experience with adaptive kernel methods [Silverman, 1986] applied to the problem of breakthrough curve reconstruction is summarized in this Appendix. [55] Adaptive methods that adjust the degree of smoothing to the local data density are considered here. Nonadaptive kernel methods have been used to reconstruct spatial variation in concentration from particle locations in conventional particle tracking methods [e.g., Bagtzoglou et al., 1992]. For retention models that produce a strong, kinetically controlled tail in the breakthrough curve (e.g., matrix diffusion), adaptive methods are preferred because these methods are able to provide needed smoothing in the tail of the breakthrough curve without obliterating details of the breakthrough curve near the peak. [56] Two adaptive kernel methods were tested. The adaptive kernel methods are well-known algorithms that have been investigated in detail [Silverman, 1986]. In these algorithms, a nearest neighbor method is first used to estimate an initial or ‘‘pilot’’ estimate. The pilot estimate is then used to calculate a variable kernel width for use in a kernel estimation method. The adaptive two-step method is
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appealing because of its capability to provide stable and smooth density estimations for a wide variety of data distributions, while maintaining sensitivity to local details. Gaussian and lognormal kernels were tested. [57] The pilot estimate for the breakthrough curve is obtained at a specified time t from a generalized nearest neighbor estimator of order k far0 ðt Þ ¼
n 1 X wi K ðt; ti ; dk Þ W i¼1
ðB1Þ
where t1, t2, , tn are the arrival times, w1, w2, , wn are the statistical weights (masses) for the individual packets, n P wi, dk is the distance from t to the k-th nearest W = i¼1
arrival time, and K is the kernel. [58] If the Gaussian kernel is used, the kernel is K(t, ti, i dk) = d1k fg(tt dk ) where f g () is the standard normal (Gaussian) PDF. If the lognormal kernel is used, K(t, ti, dk) = fln(ti; t, dk) where fln(ti; t, dk) is the lognormal PDF with geometric mean t and log-standard deviation dk. [59] Once the pilot estimate is known, the estimate for the breakthrough curve is obtained as ^far ðt Þ ¼
n X
wi K ðt; ti ; hwi Þ
ðB2Þ
i¼1
where wi = [f 0ar(ti)/g]n , g is the geometric mean of the f 0ar(ti), n is a sensitivity parameter, and h is the global bandwidth. The product hwi is a local bandwidth that adapts the kernel width according to the local data density. On the basis of theoretical arguments and practical experience, Silverman [1986] recommends the value n = 1/2 for the selectivity parameter and that the global bandwidth be calculated as h ¼ 0:9An1=5
ðB3Þ
where A = min[standard deviation, (Q75 – Q25)/1.34] applied to t in the case of the Gaussian kernel and to ln t in the case of the lognormal kernel. Here Q75 and Q25 are the 75-th and 25-th percentiles. We adopt those recommended values. In the calculation of the pilot density, we use k = 25 in the calculation of dk in equation (B1). The method is quite insensitive to the number k of nearest neighbors used in the pilot density estimation. [60] Note that the pilot estimate is properly normalized as a probability density and is used only to estimate the local bandwidth. Equation (B2) is, conversely, not normalized, and its integral represents the total amount of mass reaching the pathway outlet. [61] Both reconstruction algorithms were tested in the test simulations described in this paper. The adaptive Gaussian kernel worked well in most situations. However, it did tend to oversmooth the breakthrough curves at early times if the breakthrough curve was very broad. This oversmoothing had the unfortunate effect of causing a nonzero mass discharge to be estimated at zero or negative times. The lognormal estimator also performed quite well and always produces mass discharge of zero at nonpositive times. Thus the lognormal kernel is superior for reconstruction of the
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breakthrough curves. The results shown in this paper rely exclusively on the lognormal kernel.
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V. Cvetkovic, Royal Institute of Technology, Stockholm, Sweden. J. Mancillas, S. Painter, and O. Pensado, Center for Nuclear Waste Regulatory Analyses, Southwest Research Institute, San Antonio, TX 78228-0510, USA. (
[email protected])
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